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By adding another $2$ pebbles in line you double the area to $2$, like this:
The rule that's developing is that you keep the pebbles that are down already (not moving them to any new positions) and add as FEW pebbles as necessary to DOUBLE the PREVIOUS area, using RECTANGLES ONLY!
So, to continue, we add another three pebbles to get an area of $4$:
You could have doubled the area by doing:
But this would not obey the rule that you must add as FEW pebbles as possible each time. So this one is not allowed.
Number 6 would look like this:
Well, now it's time for you to have a go.
"It's easy,'' I hear you say. Well, that's good. But what questions can we ask about the arrangements that we are getting?
We could make a start by saying "Stand back and look at the shapes you are getting. What do you see?'' I guess you may see quite a lot of different things.
It would be good for you to do some more of this pattern. See how far you can go. You may run out of pebbles, paper or whatever you may be using. (Multilink, pegboard, elastic bands with a nail board, etc.)
Well now, what about some questions to explore?
Here are some I've thought of that look interesting:
Try to answer these, and any other questions you come up with, and perhaps put them in a kind of table/graph/spreadsheet etc.
Do let me see what you get - I'll be most interested.
Don't forget the all-important question to ask - "I wonder what would happen if I ...?''
Use this activity to introduce children to an investigation that mixes both shape and space work with number work. You could also introduce learners to this extended piece of work to help you look at perseverence and persistence.
A good introduction can be had with the whole class by making the first two or three arrangements all together. It is useful to have squared and dotted (squares) paper available whilst some pupils may benefit from using blocks (such as multilink) to represent the pebbles. You may also find it helpful to use a virtual geoboard for sharing ideas amongst the whole group.
Some pupils may produce a table or a spreadsheet of their results which would enable them to explore further.
Alternatively, Making Squares is an extension activity.
Go to More Pebbles.
Children may benefit from adult support in keeping track of where they are in their exploration. They could be helped to proceed as if it were a game.
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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?